有穷长半模偏序集上的一个定理

A theorem in a semimodular poset of finite length

本文是对参考文献[1]第二章定理14证明的补充和改进。该定理乃指: 定理14.在任何上(或下)半模的有穷长的偏序集内,Jordan-Dedekind链条件成立。 文献[1]中,考虑上半模情形,谈到已知一个连接链:γ∶a=x_0<x_1<…<x_m=b,有长m,则a,b之间每一链有长m,…,m≥2时,假设,γ'∶a=y_0<y_1<…<y_n=b为其他的连接a,b的(有穷)极大链,x_1≠y_1时(注:若x_1=y_1,则由归纳假设而证毕)存在一元u复盖x_1,y_1,….做一个连接u及b的极大链γ"(图1),但是u‖b时,γ"是不存在的(图2),本文对此给与了补充和改进。

In this paper, we suplement and improve the proof of the theorem 14 in Reference[l], chapter Ⅱ of this paper.The main results is the following proposition. Proposition, Let p be an uper semimodular poset of finite length.γ: a = x0<x1 < … <xm = b be one connected chain in p, has length m, γ': a = y0<y1<… <yn = b be another (finite) connected chain in p, connecting a and b.There exist an element u which covers x1 and y1 except in trivial case that x1 = y1 = I, If b is a maximal element in p, then u (?)b.